H. SAIBI November 11 th 2015 Outline of Lecture: Reflection, Transmission, and Refraction Diffraction The Doppler Effect The Doppler shift and relativity Shock Waves Waves encountering barriers Reflection, Transmission, and Refraction 2008 by W.H. Freeman and Company Reflection Coefficient (r)=h r /h in Transmission Coefficient ( )=h t /h in (1) Reflection and Transmission coefficients Fig.1. (a) a wave pulse traveling on a string attached to a more massive string in which the wave speed is half as large. The reflected pulse is inverted, whereas the transmitted pulse is not. (b) A wave pulse traveling on a string attached to a less massive string in which the wave speed is twice as large. In this case, the reflected pulse is not inverted. Heavier string Lighter string Fresnel relations (valid for light and sound waves) Exercise: Two Soldered Wires Two wires, different linear mass densities. Stretched under a tension F T (the tension is the same in both wires). The wave speed in the first wire is twice that in the second. A harmonic wave traveling in the first wire is incident on the junction of the wires. (A) if the amplitude of the incident wave is A, what are the amplitudes of the reflected and transmitted waves? (B) what is the ratio 2 / 1 of the mass densities of the wires? (C) what fraction of the incident average power is reflected at the junction and what fraction is transmitted? 2008 by W.H. Freeman and Company Waves encountering barriers Reflection, Transmission, and Refraction Energy conservation gives another relation between the reflection and transmission coefficients. This relation is given by: In three dimensions, a boundary between two regions of differing wave speed is a surface. The transmitted ray is bent toward or away from the normal-depending on whether the wave speed in the second medium is less or greater than that in the incident medium. 2008 by W.H. Freeman and Company Fraction of the incident power that is reflected Fraction of the transmitted Fig.2. A wave striking a boundary surface between two media in which the wave speed differs. Part of the wave is reflected and part is transmitted. The change in direction of the transmitted (refracted) ray is called refraction. (2) Waves encountering barriers Reflection, Transmission, and Refraction Total Internal Reflection: When the incident angle is greater than the critical angle, there is no refracted ray. The amount of energy reflected from a surface depends on the surface. 2008 by W.H. Freeman and Company Fig.3. Light from a source in the water is bent away from the normal when it enters the air. For angles of incidence above a critical angle c there is no transmitted ray, a condition known as total internal reflection. 2008 by W.H. Freeman and Company Figure 4. In a concert hall, a reflecting shell is placed behind the orchestra, and reflecting panels are hung from the ceiling to reflect and direct the sound back toward the listeners. Check your understanding: Balloon Hearing Aid If you place the balloon (filled with carbon dioxide) between yourself and a sound source, the sound gets louder. Why is that? The balloon is to sound as a magnifying glass is to light. 2008 by W.H. Freeman and Company Diffraction If a wavefront is partially blocked by an obstacle, the unblocked part of the wavefront bends behind the obstacle. This bending of the wavefronts is called diffraction. Almost all of the diffraction occurs for that part of the wavefront that passes within a few wavelengths of the edge of the obstacle. For the parts of the wavefront that pass farther than a few wavelengths from the edge, diffraction is negligible and the wave propagates in straight lines in the direction of the incident rays. When wavefronts encounter a barrier with an aperture (hole) only a few wavelengths across, the part of the wavefronts passing through the aperture all pass within a few wavelengths of an edge. Thus, flat wavefronts bend and spread out and become spherical or circular (Fig.5). Fig.5. Plane waves in a ripple tank meeting a barrier with an opening that is approximately one wavelength wide. Beyond the barrier are circular waves that are concentric about the opening, much as if there were a point source at the opening. Diffraction 2008 by W.H. Freeman and Company Fig.6. Comparison of particles and waves passing through a narrow opening in a barrier. (a) Transmitted particles are confined to a narrow-angle beam. (b) Transmitted waves spread out (radiate widely) from the aperture, which acts like a point source of circular waves. In contrast, for a beam of particles falling upon a barrier with an aperture, the part of the beam passing through the aperture does so with no change in the direction of the particles (Fig.6). Diffraction is one of the key characteristics that distinguish waves from particles. Diffraction 2008 by W.H. Freeman and Company Fig.7. Plane waves in a ripple tank meeting a barrier with an opening width that is large compared to. The wave continues in the forward direction, with only a small amount of spreading into the regions to either side of the opening. Although waves passing through an aperture always bend, or diffract, to some extent, the amount of diffraction depends on whether the wavelength is small or large relative to the width of the aperture. If the wavelength is greater than or equal to the width of the aperture (like in Fig.5), the diffraction effects are large, and the waves speed out as they pass through the aperture-as if the waves were originating from a point source. On the other hand, if the wavelength is small relative to the aperture, the effect of diffraction is small as shown in Fig.7. Near the edges of the aperture the wavefronts are distorted and the waves appear to bend slightly. For the most part, however, the wavefronts are not affected and the waves propagate in straight lines, much like a beam of particles. The approximation that waves propagate in straight lines in the direction of the rays with no diffraction is known as the ray approximation. Wavefronts are distorted near the edges of any obstacle blocking part of the wavefronts. By near we mean within a few wavelengths of the edges. The Doppler effect The shift in frequency caused by motion is called the Doppler effect. It occurs when a sound source is moving at speeds less than the speed of sound. The Doppler Effect Consider the source moving with speed u s (Fig.8a-b) and a stationary receiver. The source has frequency f s (and period T s =1/ f s ). The received frequency f r, the number of wave crests passing the receiver per unit time, is related to wave length and wave speed v by: A wave crest leaves the source at time t 1 and the next wave crest leaves the source at time t 2 (Fig.8c). The time between these two events is, and during this time the source and the crest leaving the source at time t 1 travel distances u s T s and vT s, respectively. Consequently, at time t 2, the distance between the source and the crest leaving at time t 1 equals the wavelength. Behind the source: In front of the source: (3) Stationary receiver u s